Mode-Selected Electron-Phonon Coupling in Superconducting Pb Nanofilms Determined from He Atom Scattering

The electron-phonon coupling (EPC) strength for each phonon mode in superconducting Pb films is measured by inelastic helium atom scattering (IHAS). This surprising ability of IHAS relies on two facts: (a) In ultrathin metal films, the EPC range exceeds the film thickness, thus enabling IHAS to detect most film phonons, even 1 nm below the surface; (b) IHAS scattering amplitudes from single phonons are shown, by first-principle arguments, to be proportional to the respective EPC strengths. Thus IHAS is the first experiment providing mode-selected EPC strengths (mode-lambda spectroscopy).

Figure 1

The experimental phonon dispersion curves of 5 (a) and 7 ML (b) Pb(111) films ($T=140\mathrm{K}$) on Cu(111), measured by IHAS at an incident beam energy of 23 meV along two symmetry directions (●), are compared with the density functional perturbation theory calculations. The phonon branches are labeled (${\alpha }_i,{\beta }_i,\dots$; $i=1,2,\dots$) following the convention of Ref. 11. Heavy red lines represent the active modes with prevalent SV polarization; broken lines in the $\overline{\Gamma }\overline{M}$ direction are shear-horizontal modes. The oblique lines are typical scan curves for incident angles of 37° (5 ML) and 37.5° (7 ML). They serve to assign the most prominent features of the energy loss spectra to the energy and parallel momentum of the phonons [11]. Panels (c) and (d) show the respective energy transfer spectra from time-of-flight measurements at the same angles ($\Delta E<0$ for phonon creation processes). The diffuse elastic scattering peak $E$ is due to a small concentration of surface defects found on even the most perfectly structured surfaces. The calculated electron-phonon coupling constants ${\lambda }_{\mathbf{Q}\nu }$ for the SV modes intersected by the scan curves in (a),(b), indicated by colored bars in (e),(f), and the corresponding inelastic spectra [(g),(h), heavy line] including both the SV- and the weaker L-mode contributions (Lorentzian components), are in good agreement with the energy transfer spectra (c),(d). The colors for SV modes are the same as in Figs. 3a, 3b.

Figure 2

The atomic core displacements of the ${\epsilon }_1$ and ${\epsilon }_2$ phonon modes at zero wave vector of a 5 monolayer lead film on a rigid substrate induce the EDOs shown in the figure. The contour lines are for equal EDO amplitudes $\pm {10}^{-n}$ atomic units ($+$ for red, $-$ for turquoise lines) and are labeled by the exponent $n$. The core displacement amplitudes are shown by vertical arrows to the right of each of the contour plots, together with the corresponding square eigenvectors. The heavy line at $-13.9\AA$ indicates the location of the rigid substrate. Although the ${\epsilon }_2$ mode is mostly localized near the surface and ${\epsilon }_1$ near the substrate layer, both modes have almost equal EDOs at a distance from the surface corresponding to the classical He-atom turning point ${\zeta }_{\mathrm{ti}}=3.5\AA$ above the topmost atomic layer, thus explaining their comparable IHAS signals. In the conventional atom-atom collision model, the IHAS intensity for the ${\epsilon }_2$ mode would be about 5 times larger than for the ${\epsilon }_1$ mode.

Figure 3

Calculated values of the mode-selected electron-phonon coupling constants ${\lambda }_{\mathbf{Q}\nu }$ for modes of quasi-SV polarization, as functions of the two-dimensional wave vector in the $\overline{\Gamma }–\overline{M}$ symmetry direction, for 5 ML (a) and 7 ML (b) $\mathrm{Pb}/\mathrm{Cu}(111)$ films. The mode labels are the same as those in Figs. 1a, 1b and the colors as in Figs. 1e, 1f.